WOLFRAM|DEMONSTRATIONS PROJECT

Cooling with a Shaped Pin Fin

​
discretization
solution
pin fin image
Biot number
1
shape factor
0.3
Contents cannot be rendered at this time; please try again later
Contents cannot be rendered at this time; please try again later
In electronic systems, a fin is a heat sink (passive heat exchanger) that cools a device by dissipating heat to the surrounding medium (e.g. air). Such a pin fin is used to maximize heat transfer to a fluid between two walls.
A pin fin has an elliptical external surface, as shown in the "pin fin image".
The walls are at a high temperature
T
w
. The fluid flowing over the pin has a free stream temperature
T
∞
. The coefficient of heat transfer between the pin wall and the surrounding medium is labeled
h
(in
W/
2
m
K
). If one introduces the dimensionless temperature
ϕ=
T-
T
∞
T
w
-
T
∞
, the governing equation is:
1
r
∂
∂r
r 
∂ϕ
∂r
+
2
∂
ϕ
∂
2
z
=0
.
The associated boundary conditions are then:
∂ϕ
∂z
z=0
=0
(axial symmetry condition),
∂ϕ
∂r
r=0
=0
(radial symmetry condition),
∂ϕ
∂r
+
h
k
ϕ
r=
r
0
(z)
=0
(continuity of heat flux at the boundary fin/surrounding air), and
ϕ(z=L)=
ϕ
w
=1
(constant temperature at the wall),
where
k (W/m K)
is the thermal conductivity of the cylindrical pin fin and
r
0
(z)=1-α
1-
2
z
.
This Demonstration plots the contours of the dimensionless temperature for a user-set value of the Biot number
Bi=
h
r
0
(1)
k
and the shape factor
α
.